Exercise 14.4 — Ogive Curves
Ogive curves and cumulative frequency graphs.
Exercise 14.4 — Statistics: Ogives (Cumulative Frequency Curves)
Class 10 Mathematics · CBSE, Telangana & Andhra Pradesh Syllabus · Chapter 14
Exercise 14.4 is the graphical wrap-up of Chapter 14, Statistics, in Class 10 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus). Instead of only calculating the median using a formula, this exercise teaches you how to draw it — by plotting a cumulative frequency curve, called an ogive, and reading the median directly off the graph.
Below you'll find a clear explanation of both types of ogive — less than type and more than type — how to convert a frequency table into each one, how to locate the median from a single ogive or from the intersection of two ogives, and complete, step-by-step solutions to all 3 problems in this exercise, each with its own plotted curve.
What Is an Ogive?
A curve that represents the cumulative frequency distribution of grouped data on a graph is called a Cumulative Frequency Curve, or an Ogive. There are two ways to build one, depending on whether you accumulate frequencies from the bottom up or from the top down.
Less Than Type Ogive
Plot the upper limit of each class (x-axis) against its cumulative frequency (y-axis) — the running total of everything below that value.
- Cumulative frequency keeps increasing
- Curve rises like an elongated "S"
- Starts near 0, ends at the total n
More Than Type Ogive
Plot the lower limit of each class (x-axis) against its cumulative frequency (y-axis) — the running total of everything at or above that value.
- Cumulative frequency keeps decreasing
- Curve falls like an upside-down "S"
- Starts at the total n, ends near 0
When both ogives are drawn on the same graph, they cross at exactly one point — the x-coordinate of that crossing point is the median of the distribution.
- Single-ogive method: Draw only the less than type ogive, locate n/2 on the y-axis, draw a horizontal line to the curve, then drop straight down to the x-axis — that x-value is the median.
- Two-ogive method: Draw both the less than and more than type ogives on the same axes. Wherever they intersect, the x-coordinate of that point is the median.
- Both methods give the same answer as the median formula — the graph is simply a visual shortcut to the same calculation.
Problem 1 — Less Than Type Ogive for Daily Income
Question: The table below shows the daily income of 50 factory workers. Convert this to a less than type cumulative frequency distribution and draw its ogive.
| Daily Income (₹) | 250–300 | 300–350 | 350–400 | 400–450 | 450–500 |
|---|---|---|---|---|---|
| No. of workers (f) | 12 | 14 | 8 | 6 | 10 |
| Daily Income (less than) | Cumulative Frequency |
|---|---|
| Less than 300 | 12 |
| Less than 350 | 12 + 14 = 26 |
| Less than 400 | 26 + 8 = 34 |
| Less than 450 | 34 + 6 = 40 |
| Less than 500 | 40 + 10 = 50 |
Plot the points (300, 12), (350, 26), (400, 34), (450, 40), and (500, 50), then join them with a smooth rising curve:
Less than type ogive for daily income — the curve rises steadily from (300, 12) up to (500, 50).
Problem 2 — Median Weight From an Ogive (Graph + Formula)
Question: During a medical check-up, the weights of 35 students were recorded as a less than type cumulative table. Draw the ogive, read off the median weight from the graph, and then verify it using the median formula.
| Weight (kg) | <38 | <40 | <42 | <44 | <46 | <48 | <50 | <52 |
|---|---|---|---|---|---|---|---|---|
| No. of students (cf) | 0 | 3 | 5 | 9 | 14 | 28 | 32 | 35 |
Plot the points (38,0), (40,3), (42,5), (44,9), (46,14), (48,28), (50,32), (52,35). Since n = 35, n/2 = 17.5. Locate 17.5 on the y-axis, move across to the curve, then drop down to the x-axis:
The dashed lines show how 17.5 on the y-axis leads to the point on the curve directly above x = 46.5 — the graphical median.
Convert the cumulative table into class-wise frequencies, then apply the standard median formula:
| Class | Frequency (f) | Cumulative Frequency (cf) |
|---|---|---|
| Below 38 | — | 0 |
| 38 – 40 | 3 − 0 = 3 | 3 |
| 40 – 42 | 5 − 3 = 2 | 5 |
| 42 – 44 | 9 − 5 = 4 | 9 |
| 44 – 46 | 14 − 9 = 5 | 14 |
| 46 – 48 (median class) | 28 − 14 = 14 | 28 |
| 48 – 50 | 32 − 28 = 4 | 32 |
| 50 – 52 | 35 − 32 = 3 | 35 |
Problem 3 — More Than Type Ogive for Wheat Yield
Question: The table below gives the wheat production yield per hectare for 100 farms. Convert this to a more than type distribution and draw its ogive.
| Yield (Qui/Hec) | 50–55 | 55–60 | 60–65 | 65–70 | 70–75 | 75–80 |
|---|---|---|---|---|---|---|
| No. of farmers (f) | 2 | 8 | 12 | 24 | 38 | 16 |
For a more than type table, start from the bottom class and accumulate upward — "more than 50" must include every farm in the data, while "more than 75" includes only the last class:
| Yield (more than) | Cumulative Frequency |
|---|---|
| More than 75 | 16 |
| More than 70 | 16 + 38 = 54 |
| More than 65 | 54 + 24 = 78 |
| More than 60 | 78 + 12 = 90 |
| More than 55 | 90 + 8 = 98 |
| More than 50 | 98 + 2 = 100 |
Plot the points (50,100), (55,98), (60,90), (65,78), (70,54), (75,16), then join them with a smooth falling curve:
More than type ogive for wheat yield — the curve falls steadily from (50,100) down to (75,16), the classic "upside-down S" shape.
Common Mistakes to Avoid
- Using the wrong class limit: Less than type ogives always use the upper limit of each class on the x-axis; more than type ogives always use the lower limit. Mixing these up gives a curve with the wrong shape.
- Forgetting the starting point: A less than type ogive should start at (lower limit of first class, 0) — e.g. (250, 0) in Problem 1 — even though this point is sometimes omitted from the data table itself.
- Drawing straight lines instead of a smooth curve: An ogive is meant to be a smooth, freehand curve through the plotted points, not a series of sharp straight-line segments.
- Misreading the y-axis scale: Always check the scale stated on the graph (e.g. "1 unit = 10 workers") before reading off values — misreading the scale is the most common source of an incorrect graphical median.
- Forgetting to halve n: When finding the median graphically, you always locate n/2 on the y-axis, not n itself — this is the single most common slip in this exercise.
- Not converting cumulative tables back to plain frequencies: As in Problem 2, when the data is already given in "less than" form, you must subtract consecutive cumulative values to recover f before applying the median formula.
Quick Reference — All 3 Answers at a Glance
| Q.No | Scenario | Key Result |
|---|---|---|
| 1 | Daily income of 50 workers | Less than type ogive through (300,12)…(500,50) |
| 2 | Weight of 35 students | Median weight = 46.5 kg (graph & formula agree) |
| 3 | Wheat yield of 100 farms | More than type ogive through (50,100)…(75,16) |
What This Exercise Prepares You For
Exercise 14.4 brings together everything from Exercise 14.3 (Median of Grouped Data) and presents it visually, completing your understanding of all three measures of central tendency taught across this chapter — mean, median, and mode.
The skill of carefully plotting points and reading values off a curve also carries forward into Coordinate Geometry, where precise plotting and reading of graphs is a core technique, and into the Probability chapter, where organising and visualising frequency data is just as important.